You really need to read up more info on 3 areas Work Permits / Work Visas and Work Authorisations. 2 of these may apply to your girlfriend. Read other posts on this site for links. But in a nutshell, yes, only skilled migrants are required currently, and it is especially hard for non-EU workers to get in at the moment due to the massive influx of available workers from the new accession States. Check out
www.oasis.gov.ie for more info.
Yes, she could work up to 20 hours a week on a Student visa, but she must register for an approved course to be accepted for a Student visa. You cannot convert to a Student visa from a Tourist visa. You must apply from outside the Republic. Check
www.justice.ie for more details on student visas.
As I said earlier, if she is applying for another on her own merits (i.e. not applying to come to you, or are your GF/Fiance) she should omit the overstay and working illegally as currently the Irish Immigration do not know these facts, nor do they know she is currently still here.
The problem may occur if she later applies to come back as your fiancé and you try to say that you knew her for X months during 2006, but she will have said that she wasnt present in Ireland during these months. Basically you are in a Catch 22 situation. You can tell the truth about her situation and risk that any future visa application will be negatively received, or you an lie, but risk getting caught out on the lies at a later stage in the immigration process.
Although it may be hard, do try to think about the long term when facing this issue, as short term solutions may end up costing you dearly in the long term.